Timeline for Weil reciprocity vs Artin reciprocity

9 events

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Apr 13, 2014 at 4:48	vote	accept	Kiu
May 9, 2012 at 20:33	comment	added	Kiu		Thanks guys for the comments. What I meant above is what Dustin also pointed out.
May 8, 2012 at 19:42	comment	added	David E Speyer		Oh I see. I was thinking we evaluate $f$ and $g$ first and then plug in, but, of course, doing it in the other order makes more sense. So Kiumars' definition is right even when $\mathrm{div}(f)$ and $\mathrm{div}(g)$ do overlap, without modification. My bad.
May 8, 2012 at 17:07	comment	added	Dustin Clausen		David - doesn't this agree with what Kiumars wrote? Since $f^ag^{-b}$ has neither a zero nor a pole at p, it can be evaluated at $p$.
May 8, 2012 at 16:12	comment	added	David E Speyer		The Weil symbol can be defined when $\mathrm{div}(f)$ and $\mathrm{div}(g)$ overlap. Let $\pi$ be a uniformizer at $p$ and write $f = f_0 \pi^b$ and $g = g_0 \pi^a$. Then $(f,g)_p = (-1)^{ab} f_0(p)^a g_0(p)^{-b}$. Exercise: This is independent of the choice of $\pi$.
May 8, 2012 at 15:13	answer	added	Dustin Clausen		timeline score: 9
May 8, 2012 at 13:55	comment	added	Chandan Singh Dalawat		Strictly speaking, the Artin symbol makes sense only for (number fields and) function fields over finite fields. But there is version of Weil reciprocity for the latter, and your question would still make sense.
May 8, 2012 at 4:34	comment	added	Chandan Singh Dalawat		Perhaps you need to assume that $\mathrm{div}(f)$ and $\mathrm{div}(g)$ are disjoint.
May 8, 2012 at 4:00	history	asked	Kiu	CC BY-SA 3.0